Soundhole diameter to box volume for maximum loudness?

[Michael Recchione]

I apologize if this has been discussed to death - I searched and didn't find anything, though. I've just built my first "legitimate" instrument, with no soundhole (it was an anglo-saxon lyre). Most of my previous builds have been cigar box guitars. (I feel like I just said "Littering" on the Group W Bench... ). I try to build them so that they sound as good and as loud as possible unplugged, which is a challenge given that they're cigar boxes...

I have read, in a couple of places (but without attribution), that, for maximum loudness, the soundhole radius should be roughly 1/4 the radius of the equivalent sphere, treating the box as a Helmholtz resonator. I've found loads of references that derive the Helmholtz resonance frequency, but I haven't seen any that show how this ratio of 1/4 for maximum loudness was derived. And every place that I've seen this ratio quoted just basically says, "I don't pretend to understand the math, but I've heard that this was true."

My questions are these: (1) Is this correct, i.e. does a ratio of 1/4 for the radii of the soundhole and the equivalent sphere really make the instrument the loudest it can be (assuming all other things held constant, and assuming the strings are tuned appropriately to excite the resonator at an appropriate frequency...), and (2) can anyone point me to a reference where this "urban legend" is derived?

Thanks in advance!

- Mike

[Alan Carruth]

Wish I could help: maybe Trevor Gore will show up...

I have to wonder if the fact that it's not a simple Helmholtz resonator changes anything? I'd bet it does. Also, maximum loudness is not everything.

[Michael Recchione]

Thanks for the reply Maximum loudness isn't everything, and yes, part of this is just intellectual curiosity. But building a cigar box guitar is a little different than building a "real guitar", because the box is a "given". Most of the time, this makes very little difference, because people mostly play the thing through an amp. But I try to go for the best (and biggest) unplugged sound that I can get. What occurred to me is that the best way to do that would be to design the instrument around the acoustics of the box, i.e. a procedure like:

(1) For the box volume, determine the soundhole size that gives maximum volume. (For the biggest sound).
(2) From the (adjusted) Helmholtz resonance for this soundhole and box, determine the optimal tuning for the strings, i.e. selected such that the Helmholtz resonant frequency is in between the fundamental frequencies of the two lowest strings, somewhat closer to the second-lowest pitch. Assuming you do the right thing with the box, this will give the "best" sound that the instrument can have.
(3) Determine the scale length that permits this tuning using reasonable strings and tensions.

My impression is that this is different from the way you'd approach a "real" guitar - in this case, the box determines the scale length, the tuning, etc. I've never approached a build in this way before, but am wondering if this would give better results than trial-and-error...

[Trevor Gore]

...maybe Trevor Gore will show up...

Like a bad penny...

Like you, I've heard the "urban myths" but have never been able to identify a source. I've looked pretty closely through Helmholtz's "On the Sensations of Tone", which is where you'd expect to find something if Helmholtz actually did the work, but couldn't find anything. But the idea is problematical for guitar design anyway, because even if such a relationship was established as "optimum" (rather than as perhaps some design guideline) it would be predicated on having a rigid bodied vessel, which a guitar (cigarbox or otherwise) certainly isn't.

If you work through the numbers, ignoring the "rigid" problem for the moment, you get numbers in the right post code, though perhaps not the right ball park.

But, even if the relationship was proven, I think you'd be missing the point if you designed specifically around that parameter. Though I've never worked it out, I wouldn't expect the air resonance of a CB guitar to be very low, so most of the bass you're hearing is probably due to the psychoacoustic "missing fundamental" phenomenon. My guess is that you'd be better off maximising the mobility of the soundboard, but if the "rules" are "don't mess with the box", you have what you have. Have a look at equivalently sized ukes and see if you can borrow ideas from there. Plenty of colour, glossy pictures around...

[G.S. Monroe]

... My guess is that you'd be better off maximising the mobility of the soundboard, but if the "rules" are "don't mess with the box", you have what you have...

Rules?

Cigar Box Guitars?

[Michael Recchione]

...maybe Trevor Gore will show up...

Like a bad penny...

Like you, I've heard the "urban myths" but have never been able to identify a source. I've looked pretty closely through Helmholtz's "On the Sensations of Tone", which is where you'd expect to find something if Helmholtz actually did the work, but couldn't find anything. But the idea is problematical for guitar design anyway, because even if such a relationship was established as "optimum" (rather than as perhaps some design guideline) it would be predicated on having a rigid bodied vessel, which a guitar (cigarbox or otherwise) certainly isn't.

If you work through the numbers, ignoring the "rigid" problem for the moment, you get numbers in the right post code, though perhaps not the right ball park.

But, even if the relationship was proven, I think you'd be missing the point if you designed specifically around that parameter. Though I've never worked it out, I wouldn't expect the air resonance of a CB guitar to be very low, so most of the bass you're hearing is probably due to the psychoacoustic "missing fundamental" phenomenon. My guess is that you'd be better off maximising the mobility of the soundboard, but if the "rules" are "don't mess with the box", you have what you have. Have a look at equivalently sized ukes and see if you can borrow ideas from there. Plenty of colour, glossy pictures around...

Well, I'm glad I'm not the only one who went off on a wild goose chase looking for the source of the "1/4 of the radius of the sphere with the same volume of the box" legend. You're probably right - this is pretty silly. My approach to date has been more ad-hoc.

I do believe that doing things like maximizing the mobility of the soundboard and sizing the box/soundhole/tuning/scale length are independent, though. No reason not to do both.

I suspect you're right - the 1 to 4 "guideline" (since you can't have rules for CBGs) is probably just an urban legend. If I have time, and this is still on my mind in a week or so, I'm going to try to test it. If I do that, I'll publish what I find here.

Thanks again for the response!

[Alan Carruth]

It occurred to me that William Allen might have said something about this in his article on air resonances in American Lutherie #1, so I dug it out. Nope. He does have a lot of interesting stuff, though, such as a discussion of how the sound hole location changes the Helmholtz resonance. Nothing's ever simple....

[Chris Paulick]

I've read where as you increase the size of the sound hole the guitar will get louder to a point and then not. Heard of some taking it to that point and when it gets past the point they will bind the soundhole to get it back to the loudest point. I don't know if it's worth all the efford though.

[Trevor Gore]

I'd be more concerned about placing the coupled main air resonant frequency when changing the sound hole diameter and looking elsewhere (e.g. higher top mobility) for loudness and projection.

[Alan Carruth]

The hole has to get pretty small to 'choke' a guitar seriously. That happened to one of my students, though. He built a reso, and had only a small sound hole as part of his decorative scheme. I thought it would be too small, and sure enough... In the end he reinforced the right side in the upper bout, and opened a port wit a handshaft grinder. He kept enlarging the port until the sound started getting worse, and then built a surround to dress it up, getting the hole back to the 'right' size.

As you enlarge the hole the sound gets more 'forward', and can become 'harsh'. Also, at some point, I believe you reach a point of diminishing returns as far as raising the 'main air pitch goes.